[论文解读] A Two Stage Generalized Block Orthogonal Matching Pursuit (TSGBOMP) Algorithm

Samrat Mukhopadhyay, Mrityunjoy Chakraborty|arXiv (Cornell University)|Aug 18, 2020

Sparse and Compressive Sensing Techniques参考文献 24被引用 5

一句话总结

本文提出了一种两阶段广义块正交匹配追踪（TSGBOMP）算法，用于在未知块边界的情况下恢复广义块稀疏信号。该文引入了一种新型伪块交错块限制等距性（PIBRIP），并建立了在该条件下TSGBOMP能以高概率保证精确恢复的恢复条件，其仿真性能显著优于传统BOMP算法。

ABSTRACT

Recovery of an unknown sparse signal from a few of its projections is the key objective of compressed sensing. Often one comes across signals that are not ordinarily sparse but are sparse blockwise. Existing block sparse recovery algorithms like BOMP make the assumption of uniform block size and known block boundaries, which are, however, not very practical in many applications. This paper addresses this problem and proposes a two step procedure, where the first stage is a coarse block location identification stage while the second stage carries out finer localization of a non-zero cluster within the window selected in the first stage. A detailed convergence analysis of the proposed algorithm is carried out by first defining the so-called pseudoblock-interleaved block RIP of the given generalized block sparse signal and then imposing upper bounds on the corresponding RIC. We also extend the analysis for complex vector as well as matrix entries where it turns out that the extension is non-trivial and requires special care. Furthermore, assuming real Gaussian sensing matrix entries, we find a lower bound on the probability that the derived recovery bounds are satisfied. The lower bound suggests that there are sets of parameters such that the derived bound is satisfied with high probability. Simulation results confirm significantly improved performance of the proposed algorithm as compared to BOMP.

研究动机与目标

• 为解决现有块稀疏恢复算法（如BOMP）需要已知块大小和边界这一局限性。
• 开发一种非贝叶斯贪婪算法，以在不依赖块划分先验知识的情况下恢复广义块稀疏信号。
• 利用一种新型伪块交错块限制等距性（PIBRIP）建立理论恢复条件。
• 在实高斯测量矩阵下推导概率恢复边界，表明在特定参数设置下满足恢复条件的概率较高。

提出的方法

• 该算法分两个阶段运行：首先，通过选择与残差相关性最大的连续列窗口，粗略识别块位置。
• 在第二阶段，对选定窗口内所有重叠的连续簇进行精细搜索，以识别最优非零块。
• 提出一种新型限制等距性变体——伪块交错块RIP（PIBRIP），用于建模广义块稀疏信号的结构。
• 理论分析推导出在保证精确恢复的块限制等距常数（BRIC）的上界。
• 该方法可扩展至复值信号和矩阵，但由于分析上的非平凡挑战，需特殊处理。
• 在实高斯测量矩阵下，推导出满足所导恢复条件的概率下界，表明在特定参数集合下具有较高成功概率。

实验结果

研究问题

• RQ1 当块边界未知时，两阶段贪婪算法是否能提升广义块稀疏信号的恢复性能？
• RQ2 在缺乏已知块划分的情况下，分析恢复所需的新型RIP条件是什么？
• RQ3 所提出的TSGBOMP算法在恢复精度和鲁棒性方面与BOMP相比如何？
• RQ4 在实高斯测量矩阵下，所导恢复边界的满足概率是多少？
• RQ5 该理论分析能否非平凡地推广至复值信号和矩阵？

主要发现

• 仿真结果表明，所提出的TSGBOMP算法在恢复性能上显著优于传统BOMP算法。
• 引入PIBRIP使得对未知块结构的广义块稀疏信号恢复能够进行严谨的理论分析。
• 在推导出的块限制等距常数（BRIC）上界下，该算法可保证精确恢复，且该上界依赖于信号参数。
• 对于实高斯测量矩阵，推导出满足恢复条件的概率下界，表明在适当参数设置下具有较高成功概率。
• 理论分析已扩展至复值信号和矩阵，尽管该扩展非平凡，需谨慎处理分析依赖关系。
• 利用生成函数和Hoeffding不等式，对伪块有效配置数量进行上界估计，从而导出信号排列可能性的紧致指数上界。